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Theorem flimss2 17927
Description: A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
flimss2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  ->  ( J  fLim  G )  C_  ( J  fLim  F ) )

Proof of Theorem flimss2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . . . . . . 7  |-  U. J  =  U. J
21flimelbas 17923 . . . . . 6  |-  ( x  e.  ( J  fLim  G )  ->  x  e.  U. J )
32adantl 453 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  x  e.  U. J )
4 simpl1 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  J  e.  (TopOn `  X ) )
5 toponuni 16917 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
64, 5syl 16 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  X  =  U. J )
73, 6eleqtrrd 2466 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  x  e.  X )
8 flimneiss 17921 . . . . . 6  |-  ( x  e.  ( J  fLim  G )  ->  ( ( nei `  J ) `  { x } ) 
C_  G )
98adantl 453 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  G )
10 simpl3 962 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  G  C_  F
)
119, 10sstrd 3303 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  ( ( nei `  J ) `  { x } ) 
C_  F )
12 simpl2 961 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  F  e.  ( Fil `  X ) )
13 elflim 17926 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
x  e.  ( J 
fLim  F )  <->  ( x  e.  X  /\  (
( nei `  J
) `  { x } )  C_  F
) ) )
144, 12, 13syl2anc 643 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  ( x  e.  ( J  fLim  F
)  <->  ( x  e.  X  /\  ( ( nei `  J ) `
 { x }
)  C_  F )
) )
157, 11, 14mpbir2and 889 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  /\  x  e.  ( J  fLim  G ) )  ->  x  e.  ( J  fLim  F ) )
1615ex 424 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  ->  ( x  e.  ( J  fLim  G
)  ->  x  e.  ( J  fLim  F ) ) )
1716ssrdv 3299 1  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
)  /\  G  C_  F
)  ->  ( J  fLim  G )  C_  ( J  fLim  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3265   {csn 3759   U.cuni 3959   ` cfv 5396  (class class class)co 6022  TopOnctopon 16884   neicnei 17086   Filcfil 17800    fLim cflim 17889
This theorem is referenced by:  flimfnfcls  17983  cnpfcf  17996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-fbas 16625  df-top 16888  df-topon 16891  df-fil 17801  df-flim 17894
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